elem.3.22
圆内接四边形的对角之和等于两直角。
本页以“圆内接四边形对角互补”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let ABCD be a circle, and let ABCD be a quadrilateral in it; I say that the opposite angles are equal to two right angles. Let AC, BD be joined.
连接AC和BD。
Then, since in any triangle the three angles are equal to two right angles, [I. 32] the three angles CAB, ABC, BCA of the triangle ABC are equal to two right angles. But the angle CAB is equal to the angle BDC, for they are in the same segment BADC; [III. 21] and the angle ACB is equal to the angle ADB, for they are in the same segment ADCB; therefore the whole angle ADC is equal to the angles BAC, ACB.
三角形ABC的三内角之和等于两直角。
Let the angle ABC be added to each; therefore the angles ABC, BAC, ACB are equal to the angles ABC, ADC. But the angles ABC, BAC, ACB are equal to two right angles; therefore the angles ABC, ADC are also equal to two right angles.
角CAB等于角BDC，角ACB等于角ADB，因此角ADC等于角BAC与角ACB之和。
Similarly we can prove that the angles BAD, DCB are also equal to two right angles.
两边加上角ABC，得角ABC与角ADC之和等于两直角；同理可证角BAD与角DCB之和也等于两直角。